The growth rate and balance of homogeneous tilings in the hyperbolic plane
The concepts of growth rate and balance are extended from tilings of the Euclidean plane \(\mathbf E\) to tilings of the hyperbolic plane \(\mathbf H\). The growth rate is defined as the following limit, if it exists: \[ \lambda = \lim_{n\rightarrow \infty} t_{n+1}/t_n, \] where \(t_n\) is the number of tiles in the \(n^{th}\) corona \(C_n\) of some initial configuration \(C_0\) in the tiling. If \(v_n\) and \(e_n\) denote the number of vertices and edges, respectively, in \(C_n\), then the tiling is called \textit{balanced} if the limits \[ \lim_{n\rightarrow \infty} v_n/t_n \text{ and } \lim_{n\rightarrow \infty} e_n/t_n \] exist and are finite. Growth rate and balance are independent of the initial configuration \(C_0\). For normal tilings of \(\mathbf E\) the growth rate is always \(\lambda = 1\). For tilings of \(H\) the growth rate, if it exists, is an algebraic number greater than 1 that depends on the tiling. An explicit formula is given for the growth rate of the tiling \([p,q]\) by \(p\)-gons, \(q\) at each vertex. Any such tiling of type \([p,q]\), \(1/p + 1/q < 1/2\), \(p \geq 4\) is balanced, as is any triangular tiling where each triangle has the same vertex degrees \([a,b,c]\), with the exception of those triangle tilings of type \([4,2m,2n]\), \(m \neq n\). Any tiling of \(\mathbf E\) for which the symmetry group is transitive on the tiles is balanced. This is not the case for tilings of \(\mathbf H\).
scientific article; zbMATH DE number 1062447
scientific article